Numerical simulations of disordered systems showed that quantum critical points, e.g. the
disorder driven Anderson metal-insulator transition can be characterized by electronic
states that are neither extended nor localized, they can be described as multifractals. The
existence of multifractal fluctuations have been seen in experiments over several decades
and local density of states (LDOS) fluctuations of multifractal nature have been
demonstrated a couple of years ago. There are a number of interesting consequences of
multifractality especially in case when the LDOS plays an important role, e.g.
Kondo-temperature, superconducting temperature or in case of dynamical and transport
properties.
Recently the notion of non-ergodic extended states has been introduced and investigated
in the case of random graphs. Its connection to multifractality is an open question to be
explored during this PhD work.